Coverage-Preserving Self-Healing in Partially Observable 3D UAV Swarm Networks via an Advection-Diffusion Graph Transformer

I. INTRODUCTION

• Severe partitioning disrupts information exchange, while existing approaches face limits in scalability, spatial preservation, global damage inference, or online efficiency.The paper motivates recovery that jointly considers connectivity, spatial distribution, and online efficiency.Source: §I · pp. 1–2
• The proposed framework combines sparse-report damage inference, a node-supernode graph, advection-diffusion learning, and scale-aware meta-learning for distributed recovery.Its composite loss balances connectivity restoration, maneuvering cost, motion consistency, inter-cluster coordination, and spatial preservation.Source: §I · p. 2

1) Hierarchical Communication Architecture:

• The communication architecture uses short-range links for local information exchange and topology construction, with a low-rate air-ground channel for post-damage coordination.Air-ground exchanges include disconnected-neighbor IDs, inferred damage information, model parameters, and recovery commands; real-time coordinates and trajectories are not transmitted.Source: 1) Hierarchical Communication… · p. 2

2) Communication Links:

• Inter-UAV A2A links determine the swarm topology, whereas A2G links support GCS control without participating in topology construction or connectivity recovery.A2A links follow a line-of-sight model and are established when received power exceeds the sensitivity threshold.Source: 2) Communication Links · p. 2

B. Residual UAV (RUAV) Graph Construction

• The residual UAV graph connects surviving UAVs whose Euclidean separation is within the effective communication radius; multiple connected clusters require cooperative repositioning.A single connected cluster retains global communication capability.Source: B. Residual UAV (RUAV) Graph… · pp. 2–3

C. Problem Formulation

• Self-healing seeks connected recovered positions while limiting mobility cost and preserving spatial distribution, but the resulting optimization is nonlinear and nonconvex.The paper reformulates it as hybrid-graph displacement prediction to address the cost of conventional online optimization and the limited cross-cluster awareness of purely local strategies.Source: C. Problem Formulation · p. 3

A. Damage-State Inference under Partial Observations

• Surviving UAVs report only disconnected-neighbor indices when a previously connected neighbor becomes unreachable, avoiding real-time coordinates and recovery trajectories.The reports identify the observable boundary around failed UAVs.Source: A. Damage-State Inference under… · p. 3
• Topological closure applies a silence-as-failure rule: a non-reporting node adjacent to a confirmed failure is classified as failed, enabling reconstruction inward from the observable damage boundary.The inference relies on boundary shielding: surviving UAVs adjacent to a failed UAV are included among reporters.Source: A. Damage-State Inference under… · pp. 3–4

B. Connectivity Detection of RUAV Graph

• BFS identifies connected clusters in O(|R_t| + |E′_t|), avoiding repeated Laplacian eigendecomposition with O(|R_t|^3) complexity.It traverses adjacent UAVs using negative off-diagonal Laplacian entries.Source: B. Connectivity Detection of RUAV Graph · p. 4

C. Air-Ground Asymmetric Online Recovery

• The GCS performs global inference and model adaptation, while surviving UAVs independently infer displacements locally using adapted parameters.Air-ground exchange is limited to IDs and model updates, reducing exposure of position-sensitive information and avoiding high-frequency centralized trajectory control.Source: C. Air-Ground Asymmetric Online Recovery · p. 4

1) From Physical Manifold to UAV Repair Graph:

• AdDiff-GT incorporates advection-diffusion dynamics into a node-supernode graph to provide a physics-guided inductive bias for displacement learning and topology recovery.Diffusion propagates repair potential across the airspace, while topology-constrained advection guides directional self-healing motion.Source: 1) From Physical Manifold to UAV… · p. 4
• The model treats feasible 3D airspace as a manifold and UAV recovery displacements as executable vectors in the local tangent space.The hybrid graph discretizes the damaged communication manifold using surviving UAVs as samples and supernodes as coarse abstractions of disconnected clusters.Source: 1) From Physical Manifold to UAV… · pp. 4–5

2) Hybrid Node-Supernode Representation:

• The hybrid representation preserves physical intra-cluster links and adds virtual cross-cluster links, UAV-to-supernode associations, and fully connected supernodes.These edge types encode intra-cluster connectivity, UAV-cluster coupling, geometric cross-cluster guidance, and global inter-cluster relationships.Source: 2) Hybrid Node-Supernode Representation · p. 5
• Shared 13D features encode geometry, cluster scale and identity, local connectivity, boundary status, and inter-cluster proximity for UAVs and supernodes.The features are designed to help identify UAVs with high potential for cross-cluster bridging.Source: 2) Hybrid Node-Supernode Representation · p. 5

B. Advection-Diffusion Graph Transformer

• AdDiff-GT processes 13D hybrid-graph node features through advection-diffusion layers and decodes them into 3D UAV displacement vectors.The graph is formed from surviving UAVs augmented with cluster-level supernodes.Source: B. Advection-Diffusion Graph Transformer · p. 5

1) Dynamics Modeling and Discretization:

• The discretized dynamics combine local propagation over observable hybrid-graph edges with attention-generated non-local coupling to model cluster-level interactions.The local operator carries information along communication links; the diffusion coupling captures non-local interactions and cluster dependencies.Source: 1) Dynamics Modeling and Discretization · pp. 5–6

2) Multi-Order Propagation and Learnable Series Approximation:

• A learnable finite-order propagation approximation replaces costly direct matrix-exponential computation with propagation bases from order zero through K_order.Features across propagation orders and attention heads are concatenated and fused through a learnable linear mapping.Source: 2) Multi-Order Propagation and… · p. 6

3) Output Mapping and Displacement Prediction:

• The decoder predicts 3D displacement vectors for surviving UAVs, while supernodes participate only in feature propagation and cluster-level structural modeling.Source: 3) Output Mapping and Displacement… · p. 7

1) Smallest Connected Distance (SCD):

• The smallest connected distance is the minimum virtual communication radius that connects the surviving graph, and Tm = dSCD − Rc measures the additional spatial scale physical movement must compensate for.dSCD is also the maximum edge weight of a minimum-bottleneck spanning tree of the complete distance-weighted graph.Source: 1) Smallest Connected Distance (SCD) · p. 7

2) Loss Function for AdDiff-GT:

• The composite loss balances connectivity restoration, maneuvering cost, motion consistency, inter-cluster coordination, and spatial preservation.Inter-cluster attraction encourages disconnected clusters to move toward one another, while the soft minimum-distance and link-spacing penalties regulate dispersion and UAV separation.Source: 2) Loss Function for AdDiff-GT · pp. 7–8

• A first-order meta-learning strategy learns a scale-specific initialization for each surviving-node scale, avoiding Hessian computation and topology-specific models.Source: 3) Meta-Learning for Fast Adaptation · p. 8
• Offline meta-learning samples disconnected support and query scenarios, adapts on support samples, and stores the resulting scale-specific meta-parameter library.Initially connected samples are excluded using BFS, and each sample is converted into a node-supernode hybrid graph.Source: 3) Meta-Learning for Fast Adaptation · p. 8
• Online, the GCS retrieves the initialization matching the inferred surviving-node scale, performs a limited gradient update, and broadcasts adapted parameters for UAV-side forward inference.Airborne nodes infer locally without backpropagation.Source: 3) Meta-Learning for Fast Adaptation · p. 8

D. Theoretical Analysis

• Under lower-bounded loss, Lipschitz-continuous gradients, and a standard step-size condition, online tuning decreases loss monotonically and converges to a first-order stationary point.The stated result concerns a locally stable self-healing state under these assumptions.Source: D. Theoretical Analysis · pp. 8–9
• Finite-step connectivity recovery follows if each disconnected state yields a persistent nonzero decrease in inter-cluster attraction potential until clusters merge.From m finite connected clusters, each effective merge reduces the cluster count by at least one.Source: D. Theoretical Analysis · p. 9

1) Scenario Configuration:

• The experiments use 3,000 partitioned scenarios with up to 200 UAVs and 5,000 scenarios with up to 400 UAVs, excluding initially connected topologies.The settings use a 120 m A2A communication radius and 1 m/s UAV speed.Source: 1) Scenario Configuration · p. 9

2) Baseline Methods:

• Comparisons cover geometric methods HERO and CEN, swarm-intelligence heuristic SIDR, and learning-based GCN-2017, CR-MGC, and GDR-TS.Source: 2) Baseline Methods · p. 9

4) Evaluation Metrics:

• Evaluation measures recovery steps, online planning time, communication cost, normalized-grid-entropy spatial dispersion, and scenario-level recovery and dispersion win rates.Higher normalized grid entropy indicates better spatial dispersion after recovery.Source: 4) Evaluation Metrics · pp. 9–10

1) Comparative Recovery Performance:

• Setting I: AdDiff-GT+Meta averaged 150.72 recovery steps with 100% connectivity; Setting II: AdDiff-GT averaged 681.44 steps with 100% connectivity.In Setting II, AdDiff-GT also achieved spatial dispersion of 0.9900 versus CEN's 0.3493.Source: 1) Comparative Recovery Performance · p. 10
• SIDR failed 2,386 scenarios, producing an all-scenario average of 823.41; in Setting II, GCN-2017, CR-MGC, and GDR-TS averaged 1152.51, 1203.48, and 1046.31 steps.GDR-TS had a 2.64% connectivity win rate, while SIDR failed in all 5,000 scenarios in Setting II.Source: 1) Comparative Recovery Performance · p. 10

2) Recovery Evolution Dynamics:

• For n_t=28, AdDiff-GT restored Ct=1 in approximately 1,170 steps; trajectories for n_t=193 and 292 also showed progressively faster cluster-count reduction.The figure plots connected-cluster count over recovery steps; Ct=1 denotes a fully connected network.Source: 2) Recovery Evolution Dynamics · p. 11, Figure 2 · p. 11

3) Computational Efficiency and Meta-Adaptation:

• In Setting II, meta-initialization reduced average planning latency from 17.6835 s to 11.1862 s (36.7%), while recovery took 715.73 versus 681.44 steps.Both variants maintained 100% connectivity; offline meta-learning loss converged stably.Source: 3) Computational Efficiency and… · p. 11

4) Communication Efficiency and Security:

• As damage rose from 50 to 150 UAVs, AdDiff-GT maintained one interaction round, unlike CR-MGC's repeated GCS-assisted optimization, but incurred higher airtime and receiver-volume costs.AdDiff-GT also required fewer recovery steps than CEN and CR-MGC.Source: 4) Communication Efficiency and Security · pp. 11–12, Figure 6 · p. 12

5) Spatial Distribution Preservation:

• Setting I dispersion was 0.9548 for AdDiff-GT and 0.9540 for AdDiff-GT+Meta; Setting II values were 0.9900 and 0.9881, versus CEN's 0.3493.SIDR's high dispersion in Setting II accompanied failure to restore connectivity in all 5,000 scenarios.Source: 5) Spatial Distribution Preservation · p. 12, Figure 7 · p. 12

6) Visual Analysis of Self-Healing Trajectories:

• With 200 damaged nodes in an N=400 network, a few boundary or bridging UAVs made larger cross-cluster movements while most nodes adjusted locally, preserving peripheral nodes.The depicted cases show asymmetric displacement fields rather than movement toward a geometric center.Source: 6) Visual Analysis of Self-Healing… · p. 12

VI. CONCLUSION

• Across 8,000 scenarios, including 5,000 large-scale cases, the conclusion reports effective connectivity recovery with favorable spatial dispersion; future work considers channel uncertainty, heterogeneous UAVs, and multi-round damage.Source: §VI · p. 12

APPENDIX A NOTATIONS AND OPERATORS

• Appendix A defines the notation and graph operators used throughout, including adjacency, degree, Laplacian, and discrete graph gradient and divergence.These operators are distinguished from their manifold counterparts.Source: Appendix A · p. 14

APPENDIX B ADDIFF-GT-BASED SELF-HEALING OF USNET

• The self-healing process proceeds through UED occurrence, local connectivity detection, and onboard forward inference with maneuver-based topology recovery.The process is illustrated in Fig. 9.Source: Appendix B · p. 14

APPENDIX C THEORETICAL ANALYSIS FOR ADDIFF-GT

• For the weighted position-space objective, the loss is non-increasing under the stated step-size condition, and online-tuning cluster points are stationary under smoothness and step-size conditions.The objective combines connectivity, displacement, smoothness, cluster coordination, and spatial-constraint terms.Source: Appendix C · pp. 14–15
• Under the effective merging condition, each recovery phase can reduce the cluster count, and at most m −1 effective merges are needed for connectivity.The process therefore achieves finite-step connectivity recovery.Source: Appendix C · pp. 15–16
• With a connected historical topology, reliable boundary reports, and a surviving GCS backhaul link, Algorithm 1 exactly recovers the failed-node set.The stated result also implies reconstruction from sparse event-triggered observations without global heartbeat polling or real-time position reporting.Source: Appendix C · p. 16
• Algorithm 2 identifies the exact connected clusters from the graph Laplacian, so the returned cluster count is one exactly when the graph is connected.The recovered neighbors correspond to negative off-diagonal Laplacian entries.Source: Appendix C · p. 16
• The protocol excludes explicit positional fields from post-damage messages, but this boundary does not rule out positional inference from external sensing or historical side information.The messages carry ID-level reports, inferred damage sets, and shared model parameters.Source: Appendix C · p. 17
• With a fixed hybrid graph and attention coupling, AdDiff-GT's multi-head propagation implements a learnable finite-order approximation of the layer-wise advection-diffusion evolution.The correspondence relies on a bounded spectral radius for convergence of the matrix-power series.Source: Appendix C · p. 17

A. Field Sampling on the Post-Damage Flight Manifold

• Surviving UAVs sample the repair-state field using features based on geometric, cluster-level, and local topological information.Cluster centroids supply additional sampling points through supernodes.Source: A. Field Sampling on the Post-Damage… · p. 17
• The hybrid graph is interpreted as discrete observations of the continuous state field at surviving UAV locations and cluster-level sampling points.UAV nodes are executable maneuvering entities, whereas supernodes aggregate cluster states and issue no movement commands.Source: A. Field Sampling on the Post-Damage… · p. 17
• The decoder produces each UAV's final displacement as a tangent-space vector in a discrete approximation of its flight manifold.Source: A. Field Sampling on the Post-Damage… · p. 17

B. Discretization of Manifold Operators on the Hybrid Graph

• Graph gradient and divergence approximate manifold state variation and flux divergence when edges encode local or structurally meaningful neighborhoods.Physical intra-cluster, virtual inter-cluster, and UAV-supernode edges jointly define the discrete neighborhoods.Source: B. Discretization of Manifold… · pp. 17–18
• Global attention supplies the non-local diffusion coupling between UAVs and supernodes across clusters, yielding graph dynamics ∂t = (C + βV − I)Z(t).Here C is the attention coupling matrix and V is the normalized transport operator.Source: B. Discretization of Manifold… · p. 18
• The normalized hybrid adjacency propagates information along physical and virtual structures, preserving local relationships while coupling inter-cluster links directionally.It serves as the graph-level surrogate for directional transport.Source: B. Discretization of Manifold… · p. 18

C. Matrix-Exponential Graph Propagation

• AdDiff-GT propagation approximates graph-level advection-diffusion state evolution rather than performing only single-step neighbor aggregation.This interpretation holds when the hybrid graph and attention coupling remain fixed within a propagation layer.Source: C. Matrix-Exponential Graph Propagation · p. 18

D. Finite-Order Multi-Head Approximation

• A finite-order polynomial expansion avoids the computationally intensive matrix exponential required for online recovery in large UAV swarms.Propagation bases encode information at different graph orders without explicitly computing the matrix exponential.Source: D. Finite-Order Multi-Head Approximation · p. 18
• The learnable output mapping combines propagation orders and attention heads, learning their contributions instead of using fixed Taylor coefficients.The 1/H factor preserves the averaging convention used in multi-head aggregation.Source: D. Finite-Order Multi-Head Approximation · pp. 18–19
• When attention coupling varies with the current state, the fixed-operator closed-form solution does not directly apply, and propagation is a layer-wise local linearization.Source: D. Finite-Order Multi-Head Approximation · p. 19
• The SCD is the minimum communication radius that makes the distance-threshold graph connected, equivalently the largest edge weight in a minimum spanning tree.Source: D. Finite-Order Multi-Head Approximation · p. 19
• The additional spatial scale Tm = dSCD − Rc provides a topology-aware reference for setting the online recovery algorithm’s displacement search range.Source: D. Finite-Order Multi-Head Approximation · p. 19
• Global attention compensation reduces propagation-depth sensitivity from exponential to polynomial under bounded input and compensation conditions.Purely local diffusion exhibits exponential sensitivity to topological perturbations with propagation depth.Source: D. Finite-Order Multi-Head Approximation · p. 20

APPENDIX D SUPPLEMENTARY EXPERIMENTAL RESULTS

• The supplementary figures show representative self-healing trajectories for an initial network of N = 400 UAVs.Source: Appendix D · p. 20
• One trajectory case has 100 damaged nodes in the N = 400 network.Source: Appendix D · p. 20
• A second trajectory case has 300 damaged nodes, and the figures also include final spatial distributions.Source: Appendix D · p. 20